In the last unit, Number Symbolism, we saw that in the ancient world
certain numbers had symbolic meaning, aside from their ordinary use for
counting or calculating.

In this unit we'll show that the plane figures, the polygons, triangles,
squares, hexagons, and so forth, were related to the numbers (three and
the triangle, for example), were thought of in a similar way, and in
fact, carried even more emotional baggage than the numbers themselves,
because they were visual. This takes us into the realm of Sacred
Geometry.

For now we'll do the polygons directly related to the Pythagoreans; the
equilateral triangle (Sacred tetractys), hexagon, triangular numbers,
and pentagram. We'll also introduce tilings, the art of covering a plane
surface with polygons.

In the last unit, Number Symbolism we saw that in the ancient
world certain numbers had symbolic meaning, aside from their ordinary
use for counting or calculating. But each number can be associated with
a plane figure, or polygon (Three and the Triangle, for
example).

In this unit we'll see that each of these polygons also had symbolic
meaning and appear in art motifs and architectural details, and some can
be classified as sacred geometry.

A polygon is a plane figure bounded by straight lines, called the
sides of the polygon.

From the Greek poly = many and gon = angle

The sides intersect at points called the vertices. The angle
between two sides is called an interior angle or vertex
angle.

Regular Polygons

A regular polygon is one in which all the sides and interior
angles are equal.

Polygons vs. Polygrams

A polygram can be drawn by connecting the vertices of a polgon.
Pentagon & Pentagram, hexagon & hexagram, octagon & octograms

Equilateral Triangle

Slide 5-2: Tablet in School of Athens, showing Tetractys

Bouleau

There are, of course, an infinite number of regular polygons, but we'll
just discuss those with sides from three to eight. In this unit we'll
cover just those with 3, 5, and 6 sides. We'll start with the
simplest of all regular polygons, the equilateral triangle.

Sacred Tetractys

The Pythagoreans were particularly interested in this polygon because
each triangular number forms an equilateral triangle. One special
triangular number is the triangular number for what they called the
decad, or ten, the sacred tetractys.

Ten is important because it is, of course, the number of fingers. The
tetractys became a symbol of the Pythagorean brotherhood. We've seen it
before in the School of Athens.

Trianglular Architectural Features

Slide 8-11: Church window in Quebec

In architecture, triangular windows are common in churches, perhaps
representing the trinity.

Slide 5-4: Irish Triskelions from Book of Durrow.Met. Museum of Art. Treasures of Early Irish Art. NY: Met. 1977

Its a design that I liked so much I used it for one of my own pieces.

Slide 5-5: Calter carving Mandala II

Calter photo

Slide 5-6: Closeup of wheel

Calter photo

Tilings

Slide 5-7: Pompeii Tiling with equilateral triangles

Calter photo

Tilings or tesselations refers to the complete covering of a
plane surface by tiles. There are all sorts of tilings, some of which
we'll cover later. For now, lets do the simplest kind, called a regular
tiling, that is, tiling with regular polygons.

This is opposed to semiregular tilings like the Getty pavement shown here.

Slide 5-8: Getty Pavement

Calter photo

The equilateral triangle is one of the three regular polygons that tile
a plane. the other two being the square and hexagon.

Hexagon & Hexagram

Slide 5-15: Plate with Star of DavidKeller, Sharon. The Jews: A Treasury of Art and Literature.
NY: Levin Assoc. 1992

Hexagonal Tilings

Our next polygon is the hexagon, closely related to the equilateral triangle

The hexagon is a favorite shape for tilings, as in these Islamic
designs, which are not regular tilings, because they use more than one shape.

But, as we saw, the hexagon is one of the three regular polygons will
make a regular tiling.

An Illusion

The hexagon is sometimes used to create the illusion of a cube by
connecting every other vertex to the center, forming three diamonds, and
shading each diamond differently.

Slide 5-10: Basket

Calter photo

Slide 5-11: Pavement, Ducal Palace, Mantua

Calter photo

The Hexagon in Nature

The hexagon is found in nature in the honeycomb, and some crystals such
as basalt, and of course, in snowflakes.

Slide 5-12: Snowflakes

Bentley, W. A. Snow Crystals. NY: Dover, 1962.

Six-Petalled Rose

The hexagon is popular in architectural decoration partly because it is
so easy to draw. In fact, these are rusty-compass constructions,
which could have been made with a forked stick.

Six circles will fit around a seventh, of the same diameter, dividing
the circumference into 6 equal parts, and the radius of a circle exactly
divides the circumference into six parts, giving a six petalled rose.

The hexagrarn can also be viewed as two overlapping Pythagorean tetractys.

Joseph Campbell writes; In the Great Seal of the U.S. there are two
of these interlocking triangles. We have thirteen points, for our
original thirteen states, and six apexes: one above, one below, andfour
to thefour quarters. The sense of this might be thalftom above or below,
orftom any point of the compass, the creative word may be heard, which
is the great thesis of democracy.
- The Power of Myth. p.27

Hexagonal Designs in Architecture

Hexagonal designs are common in ancient architecture, such as this church
window in Quebec.

Slide 5-22: Church Window in Quebec

Calter photo

This marvelous design is at Pompeii. It is made up of a central
hexagon surrounded by squares, equilateral triangles, and rhombi.

Slide: 5-23. Design at PompeiiCalter photo

Slide 5-24: Design on Pisa DuomoCalter photo

This hexagram is one of countless designs on the Duomo in Pisa.

Pentagon & Pentagram

Slide 5-26: Pentagram from grave marker

Calter photo

The Pentagram was used as used as a sign of salutaton by the
Pythagoreans, its construction supposed to have been a jealously guarded
secret. Hippocrates of Chios is reported to have been kicked out of the
group for having divulged the construction of the pentagram.

The pentagram is also called the Pentalpha, for it can be thought
of as constructed of five A's.

Euclid's Constructions of the Pentagon

Euclid gives two constructions in Book IV, as Propositions 11 & 12.
According to the translator T.L. Heath, these methods were probably
developed by the Pythagoreans.

Medieval Method of Construction

Supposedly this construction was one of the secrets of Medieval Mason's
guilds. It can be found in Bouleau p. 64.

Durer's Construction of the Pentagon

Another method of construction is given in Duret's "Instruction in
the Measurement with the Compass and Ruler of Lines, Surfaces and
Solids," 1525.

Its the same construction as given in Geometria Deutsch, a
German book of applied geometry for stonemasons and

Golden Ratios in the Pentagram and Pentagon

The pentagon and pentagram are also interesting because they are loaded
with Golden ratios, as shown in Boles p.48.

One place that the golden triangle appears is in the Penrose Tiling,
invented by Roger Penrose, in the late seventies. The curious thing
about these tilings is they use only two kinds of tiles, and will tile a plane
without repeating the pattern.

Making a Penrose Tiling
A Penrose tiling is made of two kinds of tiles, calledkites and darts.A kite is made from two
acute golden triangles and a dart from two obtuse
golden triangles, as shown above.

Slide 5-29: NCTM Cover

Conclusion

So we covered the triangle, pentagon, and hexagon, with sides 3, 5, and
6. We'll cover the square and octagon in a later unit.

Its clear that these figures, being visual, carried even more powerful
emotional baggage than the numbers they represent.

Next time we'll again talk about polygons, in particular the triangle.
But I won't waste your time with some insignificant and trivial fact
about the triangle, but will show that, according to Plato, triangles
form the basic building block of the entire universe!